Zadanie 4a
Find the common ratio r.
Geometric patterns and totals
Work with constant-ratio sequences, their finite sums and sigma notation, including growth and decay models.
Tę umiejętność sprawdzają 32 zadania wzorowane na arkuszach IB z lat 2022–2026.
Zadania sprawdzające tę umiejętność: 32 Otwórz w wyszukiwarce z filtrami
Zadanie 4b
Find the first term in the form a × 10k, with 1 ≤ a < 10 and integer k.
Zadanie 7a
Find (i) the common ratio of the geometric sequence; (ii) its first term.
Zadanie 7b
Express Gₙ in terms of n.
Zadanie 7d
Find the smallest positive integer n for which Aₙ>Gₙ.
Zadanie 9b
Show that q=r².
Zadanie 6a
(i) Show that . (ii) Hence find the mean of the eight areas in the form , where p is a positive rational number and a is a positive integer.
Zadanie 6b
Find the median area in the form cm2, with q a positive rational number.
Zadanie 8b
Now set k = −6. (i) Prove that the three terms form the start of a geometric sequence. (ii) Explain why the corresponding infinite geometric series has no finite sum.
Zadanie 8c
The already-investigated geometric case is k = −6. A second value of k also makes the sequence geometric. (i) Show that k2 + 7k + 6 = 0. (ii) Find the first three terms for the second value. (iii) State the sum S2m of its first 2m terms.
Zadanie 9a
Find the maximum height in cycle 6 of run 1.
Zadanie 9b
Find the distance travelled in run 1 immediately before contact 6.
Zadanie 7b
For Fund B, find to the nearest dollar (i) its balance after 5 years; (ii) the interest earned during those 5 years.
Zadanie 7c
(i) Write Fund B’s balance after n years in the form 6000qⁿ. (ii) Find the smallest number of complete years after which Fund B exceeds Fund A.
Zadanie 9e
The stage crosses the reference mark at T,2T,3T,… . Let uₙ be its greatest distance from the mark during (n−1)T<t<nT. These distances form a geometric sequence. Find (i) its common ratio; (ii) the total distance travelled if the model continues indefinitely.
Zadanie 2a
Find its value at the end of the first year.
Zadanie 2b
Find its value 9 years after purchase, to the nearest dollar.
Zadanie 2c
Find the least number n of complete years after which its value is below 15% of its original price.
Zadanie 6a
Find Sₙ.
Zadanie 6b
Hence prove S1 + S2 + ⋯ + Sn = [4(4n − 1) − 3n]/9.
Zadanie 9b
Show that s2 = at.
Zadanie 9c
Show that p>2.
Zadanie 9d
Suppose a=16 and s>0. List the first four terms of (i) the arithmetic sequence; (ii) the geometric sequence.
Zadanie 5b
Find h.
Zadanie 8d
Find both possible common ratios of v_n.
Zadanie 8e
Given v_99<0, find v_5.
Zadanie 8a
Find the total grant under (i) Plan A; (ii) Plan B.
Zadanie 8c
(i) Find total retained grant plus reserve at the end of month 6. (ii) Find the least number of complete months before this combined value reaches at least 165000 dollars.
Zadanie 8a
Find the total grant under (i) Plan A; (ii) Plan B.
Zadanie 8c
(i) Find total retained grant plus reserve at the end of month 6. (ii) Find the least number of complete months before this combined value reaches at least 165000 dollars.
Zadanie 8a
Suppose it is geometric. (i) Show that . (ii) For p>0 and infinite sum , find x.
Zadanie 4
A geometric sequence begins with 80 and has fourth term 121.67. Let Sₙ be the sum of its first n terms. Find the least positive integer n for which Sₙ>12000.